Distribution-Valued Solution Concepts

47 Pages Posted: 9 Jun 2010 Last revised: 3 Apr 2013

See all articles by David Wolpert

David Wolpert

Santa Fe Institute

James Bono

Microsoft Corporation

Date Written: April 2, 2013

Abstract

Under its conventional positive interpretation, game theory predicts that the mixed strategy profile of players in a particular noncooperative game will fall within some set determined by the game, e.g., the set of Nash equilibria of that game. Profiles outside of that set are implicitly assigned probability zero, and relative probabilities of profiles in that set are not given. In contrast, Bayesian rationality does not tell us to predict which state a system has by specifying a subset of its possible states. Rather it tells us to specify a posterior probability density over all those states, conditioned on all relevant information we have. So in particular, when the “state of a system” is the mixed strategy profile of the players of a game, and our information comprises the game specification, Bayesian rationality tells us to predict that profile with a posterior density over the set of all profiles, conditioned on the game specification. Here we introduce such a posterior density over profiles. Via standard Bayesian decision theory, this density fixes the unique optimal prediction of the profile of any noncooperative game, i.e., it provides a universal refinement. In addition, regulators can use such a density to make Bayes optimal choices of a mechanism, thereby fully adhering to Savage’s axioms. In particular, they can do this in strategic situations where conventional mechanism design cannot provide advice. We illustrate all of this on a Cournot duopoly game.

Keywords: Quantal Response Equilibrium, Bayesian Statistics, Mechanism Design, Cournot game, Entropic prior

JEL Classification: C02, C11, C70, C72

Suggested Citation

Wolpert, David and Bono, James, Distribution-Valued Solution Concepts (April 2, 2013). Available at SSRN: https://ssrn.com/abstract=1622463 or http://dx.doi.org/10.2139/ssrn.1622463

David Wolpert (Contact Author)

Santa Fe Institute ( email )

1399 Hyde Park Road
Santa Fe, NM 897501
United States

James Bono

Microsoft Corporation ( email )

One Microsoft Way
Redmond, WA 98052
United States

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